A MATHEMATICAL THEORY OF RANDOM MIGRATION 53 



The viifferences between a square and a circular patch inscribed in it are 

 noteworthy, indicating the marked influence of the area at the angles. Thus 

 at the centre we have only 2 per cent, as against 3 per cent., and at £ mile 

 from the centre 11 per cent, as against 18 per cent. 



As far as the above numerical investigations are to be looked upon as anything 

 but illustrations of the nature of the calculations requisite to apply the theory 

 of random migration to the mosquito clearance problem, they must be taken : 



(i) As merely an incentive to further study of the manner in which mosquitoes 

 scatter from the breeding ponds. It would seem possible, if difficult, to experimentally 

 test this by in some way marking a large number of insects, and determining the 

 nature and extent of the flight. 



(ii) As indicating that permanent sterility of the protection belt is almost 

 certainly needful. The ^ to 3 per cent, of mosquitoes at the centre of the clearance 

 amounting to 6 to 18 per cent, at £ mile distance may or may not be serious, 

 but they certainly would very soon be if they were able to breed. 



(iii) As showing that on the rough numbers taken, that a clearance belt 

 of probably \ mile round a settlement would be the minimum desirable sterile zone. 

 But it is quite possible that, when the requisite constants are better known, it 

 will be found that smaller belts will suffice. It is possibly rather an exaggerated 

 view to suppose a mosquito to make six random flights of 200 yards between 

 breeding spot and breeding spot. But certainly many insects I have noted will 

 fly with great rapidity in one flight 50, 100 or 200 yards, and these flights are 

 quite distinct from " flitters." 



■ (17) Conclusions. The present memoir suffers of course from all the defects which 

 must accompany a first attempt to develop a mathematical theory of phenomena 

 which have hitherto not been studied with this development in view. The theory 

 itself suggests hypotheses and constants which have never yet been considered. 

 How far with a broad average of environment in relation to food supply, breeding 

 places, shelter, foes, etc. is the spread of a species random ? Are any of the 

 geographical limits to plant or insect or animal life non-environmental and in 

 course of change ? If so, statistical studies of the density gradients of such species 

 for a few miles either side of the supposed boundary would form most interesting 

 work for biometricians. But, apart from this observational work, a good deal of 

 experimental inquiry might be usefully attempted with regard to the constants 

 of random scatter or flight in the cases of both seeds and insects. 



On the theoretical side there are many problems left untouched. The present 

 memoir has only opened up the outskirts of a very big field. It would be of 

 value to investigate the number of terms in the expansion in ^-functions requisite 

 to practically reproduce the graphically constructed density distributions for 

 migrations of 3, 4 or 5 flights. Our expansion to 6 terms is hardly close enough 



